FOM: set vs category as foundation

[Martin Davis]

Just want to suggest that the right question is not which leads most
appropriately into real analysis or algebraic geometry or whatever. The
right question about a foundational approach is: what sort of insight does
it yield into the outstanding foundational problems of our time? Personally
I believe these center about undecidability/independence results. We know
that any fixed foundational scheme (leading to an r.e. set of theorems) will
fail to yield proofs for some true statements asserting that certain
Diophantine equations have no integer solutions. We know that easy questions
to ask about cardinal arithmetic are left undecided by existing foundations.

Which of the proposed foundational approaches is more likely to help in
understanding/resolving this?

Martin